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One of the seven indeterminate forms in mathematics is $0^0$ (Refer :) but as a convention its value is taken as $1$ (Refer).

How to resolve this ambiguity ? Any help is much appreciated.

Eureka
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  • in calculus, it's indeterminate; but in algebra and combinatorics, it's sometimes taken to be $1$. The contexts are different. It's like $1 + 2 + 3 ... $ is typically $\infty$ but under analytic continuation it's instead $-\frac{1}{12}$. But it doesn't mean that infinity is equal to a negative fraction, instead the way we interpret the expression changed. – okzoomer Jan 02 '22 at 02:24
  • “Indeterminate” and “undefined” are not the same thing. – Thomas Andrews Jan 02 '22 at 05:38
  • @ThomasAndrews, but knowing that $0^0 = 1$ makes it "determinate" right ? – Eureka Jan 02 '22 at 05:50
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    No, it makes it defined. As a form, it is indeterminate. As an expression. It is defined. That’s because the “form” is about limits. So a limit is of the form $0^0$ if it can be written as $\lim f(x)^{g(x)}$ where $\lim f(x)=\lim g(x)=0.$ In many cases, we can determine a limit based on its form. A limit of the form $4/2$ will be $2.$ A limit of the form $0^0$ is unknown, base on form alone, hence the form is indeterminate. That says nothing about whether we define a value for $0^0.$ We could define a value for $\frac00,$ too, and the form $\frac00$ would still be indeterminate. – Thomas Andrews Jan 02 '22 at 07:31
  • (Of course, the reason we don’t define a value for $\frac00$ is there isn’t a value which is generally useful.) – Thomas Andrews Jan 02 '22 at 07:34

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"Indeterminate form" refers primarily to the limiting behaviour of functions $f(x), g(x)$. So $0^0$ is an "indeterminate form" in the sense that the limit of $f(x)^{g(x)}$ cannot be determined by simply knowing that $f(x), g(x) \to 0$.

The debate regarding the value of $0^0$ as its own mathematical expression — not an indeterminate form or limiting value, as described above, but rather unambiguously the number zero raised to the zero-th power — is a different matter. The standard convention is that $0^0=1$, essentially since this makes combinatorics and power series work like they are supposed to.

MathematicsStudent1122
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